Question: Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{t^3 - 10t^2 + 25t}{t^3 + 4t^2 - 45t}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ k = \dfrac {t(t^2 - 10t + 25)} {t(t^2 + 4t - 45)} $ $ k = \dfrac{t}{t} \cdot \dfrac{t^2 - 10t + 25}{t^2 + 4t - 45} $ Simplify: $ k = \dfrac{t^2 - 10t + 25}{t^2 + 4t - 45}$ Since we are dividing by $t$ , we must remember that $t \neq 0$ Next factor the numerator and denominator. $ k = \dfrac{(t - 5)(t - 5)}{(t - 5)(t + 9)}$ Assuming $t \neq 5$ , we can cancel the $t - 5$ $ k = \dfrac{t - 5}{t + 9}$ Therefore: $ k = \dfrac{ t - 5 }{ t + 9 }$, $t \neq 5$, $t \neq 0$